| L(s) = 1 | + 8·7-s − 12·17-s + 16·23-s − 8·25-s + 8·31-s − 4·41-s − 16·47-s + 34·49-s + 20·73-s + 24·79-s + 32·89-s + 16·97-s − 8·103-s − 32·113-s − 96·119-s + 10·121-s + 127-s + 131-s + 137-s + 139-s + 149-s + 151-s + 157-s + 128·161-s + 163-s + 167-s − 8·169-s + ⋯ |
| L(s) = 1 | + 3.02·7-s − 2.91·17-s + 3.33·23-s − 8/5·25-s + 1.43·31-s − 0.624·41-s − 2.33·47-s + 34/7·49-s + 2.34·73-s + 2.70·79-s + 3.39·89-s + 1.62·97-s − 0.788·103-s − 3.01·113-s − 8.80·119-s + 0.909·121-s + 0.0887·127-s + 0.0873·131-s + 0.0854·137-s + 0.0848·139-s + 0.0819·149-s + 0.0813·151-s + 0.0798·157-s + 10.0·161-s + 0.0783·163-s + 0.0773·167-s − 0.615·169-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 84934656 ^{s/2} \, \Gamma_{\C}(s)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 84934656 ^{s/2} \, \Gamma_{\C}(s+1/2)^{2} \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(5.233878867\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.233878867\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $\Gal(F_p)$ | $F_p(T)$ |
|---|
| bad | 2 | | \( 1 \) |
| 3 | | \( 1 \) |
| good | 5 | $C_2^2$ | \( 1 + 8 T^{2} + p^{2} T^{4} \) |
| 7 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 11 | $C_2^2$ | \( 1 - 10 T^{2} + p^{2} T^{4} \) |
| 13 | $C_2^2$ | \( 1 + 8 T^{2} + p^{2} T^{4} \) |
| 17 | $C_2$ | \( ( 1 + 6 T + p T^{2} )^{2} \) |
| 19 | $C_2^2$ | \( 1 + 6 T^{2} + p^{2} T^{4} \) |
| 23 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
| 29 | $C_2^2$ | \( 1 + 40 T^{2} + p^{2} T^{4} \) |
| 31 | $C_2$ | \( ( 1 - 4 T + p T^{2} )^{2} \) |
| 37 | $C_2^2$ | \( 1 + 72 T^{2} + p^{2} T^{4} \) |
| 41 | $C_2$ | \( ( 1 + 2 T + p T^{2} )^{2} \) |
| 43 | $C_2^2$ | \( 1 + 54 T^{2} + p^{2} T^{4} \) |
| 47 | $C_2$ | \( ( 1 + 8 T + p T^{2} )^{2} \) |
| 53 | $C_2^2$ | \( 1 + 8 T^{2} + p^{2} T^{4} \) |
| 59 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 61 | $C_2^2$ | \( 1 + 104 T^{2} + p^{2} T^{4} \) |
| 67 | $C_2^2$ | \( 1 + 6 T^{2} + p^{2} T^{4} \) |
| 71 | $C_2$ | \( ( 1 + p T^{2} )^{2} \) |
| 73 | $C_2$ | \( ( 1 - 10 T + p T^{2} )^{2} \) |
| 79 | $C_2$ | \( ( 1 - 12 T + p T^{2} )^{2} \) |
| 83 | $C_2^2$ | \( 1 + 134 T^{2} + p^{2} T^{4} \) |
| 89 | $C_2$ | \( ( 1 - 16 T + p T^{2} )^{2} \) |
| 97 | $C_2$ | \( ( 1 - 8 T + p T^{2} )^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{4} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−7.891804870130875840848646473520, −7.83755570287223920399442511480, −7.22679572171406648156939886966, −6.74472797001150349431123940884, −6.51849290520016456220239335638, −6.51652409041043753850223347652, −5.72066994805876134102740261380, −5.18099467764377355964763643297, −4.99352759546942354614067172206, −4.85095349484092413470050416117, −4.58611259194452573823520787842, −4.26175440151506504749363871199, −3.66497928552278998574062224136, −3.32642231974352204512673566878, −2.59830341567481212214609881429, −2.31949074983506726454665680792, −1.83280724864054588998172517449, −1.72020307215401050376847078255, −0.997325790249093817029402849287, −0.56907382122675692397100036096,
0.56907382122675692397100036096, 0.997325790249093817029402849287, 1.72020307215401050376847078255, 1.83280724864054588998172517449, 2.31949074983506726454665680792, 2.59830341567481212214609881429, 3.32642231974352204512673566878, 3.66497928552278998574062224136, 4.26175440151506504749363871199, 4.58611259194452573823520787842, 4.85095349484092413470050416117, 4.99352759546942354614067172206, 5.18099467764377355964763643297, 5.72066994805876134102740261380, 6.51652409041043753850223347652, 6.51849290520016456220239335638, 6.74472797001150349431123940884, 7.22679572171406648156939886966, 7.83755570287223920399442511480, 7.891804870130875840848646473520
